Syllabus for
Master of Science (Statistics)
Academic Year  (2021)

Master of Science in Statistics at CHRIST (Deemed to be University) offers the students an amalgam of knowledge on theoretical and applied statistics on a broader spectrum. Further, it intends to impart awareness of the importance of the conceptual framework of statistics across diversified fields and provide practical training on statistical methods for carrying out data analysis using sophisticated programming languages and statistical softwares such as R, Python, SPSS, EXCEL, etc. The course curriculum has been designed in such a way to cater for the needs of stakeholders to get placements in industries and institutions on successful completion of the course and to provide those ample skills and opportunities to meet the challenges at the national level competitive examinations like CSIR NET in Mathematical Science, SET, Indian Statistical Service (ISS) etc.

Programme Outcome/Programme Learning Goals/Programme Learning Outcome:

PO2: Identify the need and scope of Interdisciplinary research.

PO3: Enhance research culture and uphold scientific integrity and objectivity

PO4: Understand the professional, ethical and social responsibilities

PO5: Understand the importance and the judicious use of technology for the sustainability of the environment

PO6: Enhance disciplinary competency, employability and leadership skills

Programme Specific Outcome:

PSO2: Demonstrate the execution of statistical experiments or investigations, analyse and interpret using appropriate statistical methods, including statistical software and report the findings of experiments or studies accurately.

PSO3: Demonstrate acquaintance with contemporary trends in industrial/research settings and innovate novel solutions to existing problems.

CIA - 50%

ESE - 50%

Probability is a measure of uncertainty and forms the foundation of statistical methods. This course makes students use measure-theoretic and analytical techniques for understanding probability concepts.  

Sets – functions - Sigma field – Measurable space – Sample space – Measure – Probability as a measure - Inverse function - Measurable functions – Random variable - Induced probability space - Distribution function of a random variable: definition and properties.

Expectation and moments: Definition and properties – Probability generating function - Moment generating functions - Moment inequalities: Markov’s, Chebychev’s, Holder, Jenson and basic inequalities - Characteristic function and properties (idea and statement only).

Random vectors – joint distribution function – joint moments - Conditional probabilities - Randon-Nikodym Theorem (Statement only) - Bayes’ theorem – conditional distributions – independence - Conditional expectation and its properties

Modes of convergence: Convergence in probability, in distribution, in rth mean, almost sure convergence and their inter-relationships - Convergence theorem for expectation

Law of large numbers - Convergence of series of independent random variables - Weak law of large numbers (Kninchine’s and Kolmogorov’s) - Kolmogorov’s strong law of large numbers - Central limit theorems for i.i.d random variables: Lindberg-Levy and Liaponov’s CLT.

Probability distributions are used in many real-life phenomena. This course makes students understand different probability distributions and model real-life problems using them.

Modified power series family and properties - Binomial - Negative binomial, Logarithmic series and Lagrangian distributions and their properties as special cases of the results from modified power series family - hypergeometric distribution and its properties.

Pearsonian system of distributions - Beta, Gamma, Pareto and Normal as special cases of the Pearson family and their properties - Exponential family of distributions.

Sampling distributions of the mean and variance from normal population - independence of mean and variance - chi-square, students t and F distribution and their non-central forms - Order statistics and their distributions.

Bivariate Poisson, Multinomial distribution - Multivariate normal (definition only) - bivariate exponential distribution of Gumbel - Marshall and Olkin distribution - Dirichlet distribution.

Quadratic forms in normal variables: distribution and properties - Cochran’ theorem: applications.

This course is offered to make students understand the critical aspects of matrix theory and linear models used in different areas of statistics such as regression analysis, multivariate analysis, design of experiments and stochastic processes.

Matrix operations - Linear equations - row reduced and echelon form -  Homogenous system of equations -  Linear dependence

Vectors - Operations on vector space - subspace - nullspace and column space - Linearly independent sets - spanning set - bases - dimension - rank - change of basis.

Algebra of linear transformations - Matrix representations -  rank nullity theorem - determinants - eigenvalues and eigenvectors -  Cayley-Hamilton theorem -  Jordan canonical forms - orthogonalisation process -  orthonormal basis.

Reduction and classification of quadratic forms -  Special matrices: symmetric matrices - positive definite matrices - idempotent and projection matrices - stochastic matrices -  Gramian matrices - dispersion matrices

Fitting the model - ordinary least squares -  estimability of parametric functions -  Gauss – Markov theorem -  applications: regression model - analysis of variance.

To acquaint students with different methodologies in statistical research and to make them prepare scientific articles using LaTeX.

Objectives - Motivation - Utility - Concept of theory - empiricism - deductive and inductive theory - Characteristics of the scientific method - Understanding the language of research - Concept - Construct - Definition - Variable - Research Process  Problem Identification & Formulation - Research Question – Investigation Question - Logic & Importance

Principles of mathematical writing - LaTeX: installing packages and editor, preparing title page - mathematical expressions - tables - importing graphics - bibliography - writing a research paper - survey article - thesis writing - Beamer: preparing presentations

ESE - 50%

This course aims to impart the concepts of survey sampling theory and the analysis of complex surveys, including methods of sample selection, estimation, sampling variance, standard error of estimation in a finite population, development of sampling theory for use in sample survey problems and sources of errors in surveys.

Sampling vs census, simple random sampling: with (SRS) and without replacement (SRSWOR) of units, estimators of mean, total and variance, determination of sample size, sampling for proportions, Stratified sampling scheme: estimation and allocation of sample size, comparison with simple random sampling schemes. 

Lab Exercises:

1. Drawing samples with SRSWR and SRSWOR and estimation of parameters

2. Estimation of parameters using a sample of proportions

3. Drawing stratified sample and estimation of parameters

Bias and mean square error, estimation of variance, confidence interval, comparison with mean per unit estimator, optimum property of ratio estimator, unbiased ratio type estimator, ratio estimator in stratified random sampling, Difference estimator and Regression estimator:- Difference estimator, regression estimator, comparison of regression estimator with mean per unit and ratio estimator, regression estimator in stratified random sampling.

Lab Exercises:

4. Estimation using ratio estimator

5. Estimation using regression estimator

6. Ratio estimator and regression estimator in stratified sampling

With and without replacement sampling schemes: PPS and PPSWR schemes, Selection of samples, estimators: ordered and unordered estimators. Πps sampling schemes.

Lab Exercises:

7. Exercise on the PPS scheme

8. Exercise on the PPSWR scheme

9. Exercise on Πps sampling scheme

Systematic sampling scheme: estimation of population mean and variance, comparison of systematic sampling with SRS and stratified random sampling, circular systematic sampling, Cluster sampling: estimation of population mean, estimation of efficiency by a cluster sample, variance function, determination of optimum cluster size, Multistage sampling: estimation population total with SRS sampling at both stages, multiphase sampling (outline only), quota sampling, network sampling; Adaptive sampling: introduction and estimators under adaptive sampling. Introduction to small area estimation.

Lab Exercises:

9. Exercise on the systematic sampling scheme

10. Exercise on cluster sampling

11. Exercise on multi-stage sampling

12. Exercise on small area estimation

Sampling and non-sampling errors, the effect of unit nonresponse in the estimate, procedures for unit nonresponse

Lab Exercises:

13. Exercise on the sensitivity of efficiency due to sampling errors

14. Procedures for non-response

2. Singh, D. and Chaudharay, F.S. (2018) Theory and Analysis of Sample Survey Designs, New Age International.

2. Singh, S. (2003). Advanced Sampling: Theory and Practice. Kluwer.

3. Des Raj and Chandhok, P. (2013) Sampling Theory, McGraw Hill. 

4. Mukhopadhay, P (2009) Theory and methods of survey sampling, Second edition, PHI Learning Pvt Ltd., New Delhi.

5. Sampath, S. (2005) Sampling theory and methods, Alpha Science International Ltd., India.

6. Lumley, T. (2011). Complex surveys: a guide to analysis using R. John Wiley & Sons.

ESE - 50%

The programming skill in R helps students to perform statistical computations with ease. This course equips students with knowledge of R programming to develop statistical models for real-world problems 

R and R studio - Variables - Functions - Vectors - Expressions and assignments - Logical expressions - Matrices - The workspace - R markdown.

Practical Assignments:

1. Demonstrate variables and functions in R

2. Creating vectors and matrices and associated operations in R

3. Logical and arithmetic operations in R

Loops: if, for, while - Program flow - Basic debugging  - Good programming habits -  Input and outputs: Input from a file and output to a file 

Practical Assignments:

4. Illustration of control structures: if, else, for

5. Illustration of control structures: while, repeat, break, next and ifelse

Functions - Optional arguments and default values - Vector-based programming using functions - Recursive programming - Debugging functions - Sophisticated data structures - Factors -Dataframes - Lists - The apply family.

Practical Assignments:

7. Creating user-defined functions and doing vector-based programming

8. Creating lists and data frames and associated operations

9. Demonstration of recursive functions, apply functions in R

Visualising data - Graphical summaries of data: Bar chart, Pie chart, Histogram, Box-plot, Stem and leaf plot, Frequency table - Plotting of probability distributions and sampling distributions - P-P plot - Q-Q Plot  - ggplot2 - lattice – 3D plots,  -  par -graphical augmentation.

Practical Assignments:

10. Visualization of univariate data

11. Visualization of numerical variables in R using ‘base R’, ‘ggplot2’ and ‘lattice 3D’ packages

12. Contingency tables and visualization of categorical variables using ‘base R’, ‘ggplot2’ and ‘lattice 3D’  packages

13. Construction  of probability plots and quantile plots in R

Simulating iid uniform samples - Congruential generators - Seeding - Simulating discrete random variables - Inversion method for continuous random variables -  Rejection method - generation of normal variates: Rejection with exponential envelope, Box-Muller algorithm.

Practical Assignments:

14.  Simulation of discrete variables in R

15. Simulation of continuous variables- inversion method, rejection method

2.Matloff, N. (2016). The art of R programming: A tour of statistical software design. No Starch Press.

2.Chambers, J. M. (2008). Software for Data Analysis-Programming with R. Springer-Verlag, New York.

ESE - 50%

To provide a strong mathematical and conceptual foundation in the methods of parametric estimation and their properties.

Sufficiency - factorisation theorem - minimal sufficiency - exponential family and completeness - Ancillary statistics and Basu's theorem

UMVUE - Fisher Information and Cramer-Rao inequality - Chapman-Robbin’s and Bhattacharya bounds - Rao-Blackwell theorem - Lehman-Scheffe theorem - Unbiased estimation

Consistency - Weak and strong consistency - Marginal and joint consistent estimators - CAN estimators - equivariance - Pitman estimators

Methods of moments - Minimum chi square and its modification, Least square estimation, Maximum likelihood, Properties of maximum likelihood estimators, Cramer-Huzurbazar Theorem, Likelihood equation - multiple roots, Iterative methods, EM Algorithm.

Large sample confidence interval - shortest length confidence interval - Methods of finding confidence interval: Inversion of the test statistic, pivotal quantities, pivoting CDF- evaluation of confidence interval: size and coverage probability, loss function and test function optimality.

2.Silvey, S. D. (2017). Statistical inference. Routledge.

3.Trosset, M. W. (2009). An introduction to statistical inference and its applications with R. Chapman and Hall/CRC.

4.Dixit, U. J. (2016). Examples in parametric inference with R, Springer.

5.Lehmann, E. L., & Casella, G. (2006). Theory of point estimation, 2nd Ed. Springer.

6.Robert, C., & Casella, G. (2013). Monte Carlo statistical methods. Springer

To equip the students with theoretical and practical knowledge of stochastic models which are used in economics, life sciences, engineering etc.

A sequence of random variables - definition and classification of the stochastic process - autoregressive processes and Strict Sense and Wide Sense stationary processes.

Markov Chains: Definition, Examples - Transition probability matrix -  Chapman-Kolmogorv equation - classification of states - limiting and stationary distributions - ergodicity - discrete renewal equation and basic limit theorem - Absorption probabilities - Criteria for recurrence - Generic application: hidden Markov models.

Transition probability function - Kolmogorov differential equations - Poisson process: homogenous process, inter-arrival time distribution, compound process - Birth and death process - Service applications: Queuing models- Markovian models.

Galton-Watson branching processes - Generating function - Extinction probabilities - Continuous-time branching processes - Extinction probabilities - Branching processes with general variable lifetime.

Renewal equation - Renewal theorem - Generalisations and variations of renewal processes -  Brownian motion - Introduction to Markov renewal processes.

2.S. M. Ross (2014). Introduction to Probability Models. Elsevier.

2.J. Medhi (2009) Stochastic Processes, 3rd Edition, New Age International.

3.Dobrow, R.P. (2016), Introduction to Stochastic Processes with R, Wiley Eastern.

4.Cinlar, E. (2013). Introduction to stochastic processes. Courier Corporation.

Categorical data analysis deals with the study of information captured through expressions or verbal forms. This course equips the students with the theory and methods to analyse and categorical responses.

Categorical response data - Probability distributions for categorical data - Statistical inference for discrete data

Probability structure for contingency tables - Comparing proportions with 2x2 tables - The odds ratio - relative risk - Tests for independence - Association of IXJ tables

Components of a generalised linear model - GLM for binary and count data - Statistical inference and model checking - Fitting GLMs

Interpreting the logistic regression model - Inference for logistic regression -  Logistic regression with categorical predictors - Multiple logistic regression - Summarising effects - Building and applying logistic regression models - Multicategory logit models

Loglinear models for two-way and three-way tables - Inference for Loglinear models - the log-linear-logistic connection - Models for matched pairs: Comparing dependent proportions,  Logistic regression for matched pairs - Comparing margins of square contingency tables - symmetry issues

2.       Agresti, A. (2010). Analysis of ordinal categorical data. John Wiley & Sons.

2.      Stokes, M. E., Davis, C. S., & Koch, G. G. (2012). Categorical data analysis using SAS. SAS Institute.

3.      Agresti, A. (2018). An introduction to categorical data analysis. John Wiley & Sons.

4.      Bilder, C. R., & Loughin, T. M. (2014). Analysis of categorical data with R. Chapman and Hall/CRC.

Regression models are mainly used in establishing a relationship among variables and predicting future values. It got applications in various domain such as finance, life science, management, psychology, etc. This course is designed to impart the knowledge of statistical model building using regression technique.

Linear Regression Model: Simple and multiple -  Least squares estimation -  Properties of the estimators - Maximum likelihood estimation -  Estimation with linear restrictions -  Hypothesis testing -  confidence intervals.

Practical Assignments:

1.     Build a simple linear model and interpret the data.

2.     Construct confidence interval for the simple linear model

3.     Build a multiple linear models and estimate its parameters.

4.     Construct confidence interval for multiple linear model

Residual analysis -  Departures from underlying assumptions -  Effect of outliers - Collinearity - Nonconstant variance and serial correlation -  Departures from normality -  Diagnostics and remedies. 

Practical Assignments:

5.     Carry out residual analysis and validate the model assumptions.

6.     Construct residual plots for checking outliers, leverage points and influential points.

7. Checking the assumption of homoscedasticity and its remedial measures 

8. Detecting multicollinearity and its remedial measures

selection of input variables and model selection -  Methods of obtaining the best fit - stepwise regression -  Forward selection and backward elimination

Practical Assignments:

9.     Selecting the best model using step wise regression.

10.     Selecting the best model using the forward and backward selection procedure.

Introduction to general non-linear regression - least-squares in non-linear case - estimating the parameters of a non-linear system - reparametrization of the model -  Non-linear growth models 

Practical Assignments:

11.Estimate parameters in non-linear models using the least square procedure

Linear absolute deviation regression - M estimators -  robust regression with rank residuals -  resampling procedures for regression models, methods and its properties (without proof) -  Jackknife techniques and least-squares approach based on M-estimators.

Practical Assignments:

12.     Illustrate resampling procedures in regression models.

13.     Build a regression model with robust regression procedures.

2. Draper, N. R., & Smith, H. (2014). Applied regression analysis. 3rd edition. John Wiley & Sons.

3. Montgomery, D. C., Peck, E. A., & Vining, G. G. (2021). Introduction to linear regression analysis. John Wiley & Sons.

2. Keith, T. Z. (2014). Multiple regression and beyond: An introduction to multiple regression and structural equation modelling. Routledge.

3. Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.

4. Fox, J., & Weisberg, S. (2018). An R companion to applied regression. Sage publications.

ESE - 50%

Python is a generic programming language that is extensively used in data science. This course equips students with programming skill in Python and associated statistical libraries and to apply in data analysis

Installing Python - basic syntax - interactive shell - editing, saving  and  running  a script. The concept of data types -  variables -  assignments - mutable type - immutable types - arithmetic operators and expressions -  comments in the program - understanding error messages - Control statements - operators.

Practical Assignments:

1.  Lab exercise on data types

2.  Lab exercise on arithmetic operators and expressions 

3. Lab exercise on Control statements

Introduction to functions - inbuilt and user defined functions - functions with arguments and return values - formal vs actual arguments - named arguments - Recursive functions -  Lambda function - OOP Concepts -  classes - objects - attributes and methods - defining classes - inheritance -  polymorphism.

Practical Assignments:

4.  Lab exercise on inbuilt and user-defined functions

5.  Lab exercise on Recursive and Lambda function 

6. Lab exercise on OOP Concepts.

Introduction to Pandas - Pandas data series - Pandas data frames - data handling -  grouping - Descriptive statistical analysis and Graphical representation.

Practical Assignments:

7.  Lab exercise on Pandas data series, frame, handling and grouping

8.  Lab exercise on statistical analysis 

Hypothesis testing - data modelling - linear regression models -  logistic regression model.

Practical Assignments:

9.  Lab exercise on Hypothesis testing

10.  Lab exercise on regression modelling

Line graph - Bar chart - Pie chart - Heat map - Histogram - Density plot - Cumulative frequencies - Error bars - Scatter plot - 3D plot.

Practical Assignments:

11.  Lab exercise on graphical and diagrammatic representation.

12.  Lab exercise on the density plot

13.  Lab exercise on scatter and 3D plot

2.Haslwanter, T. (2016). An Introduction to Statistics with Python. Springer International Publishing.

2.Anthony, F. (2015). Mastering pandas. Packt Publishing Ltd.

ESE - 50%

This course provides a strong foundation for data science and the application area related to it and caters for the underlying core concepts and emerging technologies in data science.

Introduction – Big Data and Data Science  – Data science Hype – Getting Past the Hype – The Current Landscape – Role of Data Scientist – Exploratory Data Analysis –  Data Science Process Overview – Defining goals – Retrieving data – Data preparation – Data exploration – Data modelling – Presentation. Problems in handling large data – General techniques for handling large data – Big Data and its importance, Four Vs, Drivers for Big data, Big data analytics, Big data applications, Algorithms using map-reduce, Matrix-Vector Multiplication by Map Reduce. Steps in big data – Distributing data storage and processing with Frameworks – Data science ethics – valuing different aspects of privacy – The five C’s of data.

 Practical Assignments

 1. Lab exercise for feature engineering

   2. Lab exercise for big data processing

Machine learning – Modeling Process – Training model – Validating model – predicting new observations – Supervised learning algorithms – Unsupervised learning algorithms. Introduction to deep learning – Deep Feed Forward networks – Regularization – Optimization of deep learning – Convolutional networks – Recurrent and recursive nets – applications of deep learning.

Practical Assignments:

 3.  Lab exercise on Linear and Logistic discrimination 

4.  Lab exercise on K means clustering and Hierarchical clustering

Concept and Overview of DBMS, Data Models, Database Languages, Database Administrator, Database Users, Three Schema architecture of DBMS. Basic concepts, Design Issues, Mapping Constraints, Keys, Entity-Relationship Diagram, Weak Entity Sets, Functional Dependency, Different anomalies in designing a Database, Normalization: using functional dependencies, 1NF, 2NF, 3NF and Boyce-Codd Normal Form

Practical Assignments:

5. Lab Exercise on Database Design

6. Top-Down Approach

7. Bottom-up Approach 

SQL Basic Structure - DDL, DML, DCL-Integrity Constraints - Domain Constraints, Entity Constraints - Referential Integrity Constraints, Concept of Set operations, Joins, Aggregate Functions, Null Values, , assertions, views, Nested Subqueries – procedural extensions – stored procedures – functions- cursors – Intelligent databases – ECA rule – Data Integration – ETL Process

 Practical Assignments:

8. Lab Exercise on SQL

9. Lab Exercise on PL/SQL

10. Lab Exercise on ETL

Data Warehousing - Defining Feature – Data warehouses and data marts –Metadata in the data warehouse – Data design and Data preparation - Dimensional Modeling - Principles of dimensional modelling – The star schema – star schema keys – Advantages of the star schema – Updates to the dimension tables – The snowflake schema – Aggregate fact tables – Families Oo Stars – MDX queries – Reporting services.

Practical Assignments:

11. Lab Exercise on Analysis Services

12. Lab Exercise on Reporting Services

2. Thomas Cannolly and Carolyn Begg, (2007), Database Systems, A Practical Approach to Design, Implementation and Management”, 3rd Edition, Pearson Education. 

2. D J Patil, Hilary Mason, Mike Loukides, (2018), Ethics and Data Science, O’Reilly.

3. LiorRokach and OdedMaimon, (2010), Data Mining and Knowledge Discovery Handbook. 

ESE - 50%

This course will provide an introduction to the principles and methods for the analysis of time-to-event data. This type of data occurs extensively in both observational and experimental biomedical and public health studies.

The hazard and survival functions - Mean residual life function - competing risk - right,left and interval censoring, truncation - likelihood for censored and truncated data - Parametric and non-parametric estimation in truncated and censored cases.

Practical Assignments:

1.Lab exercise on the parametric estimation of left and right-censored data

2.Lab exercise on the parametric estimation of truncated data

3.Lab exercise on the non-parametric estimation of censored and truncated data

Parametric forms and the distribution of log time - The exponential - Weibull - Gompertz - Gamma - Generalized Gamma - Coale-McNeil - and generalized F distributions - The U.S. life table - Approaches to modelling the effects of covariates - Parametric families - Proportional hazards models (PH) - Accelerated failure time models (AFT) - The intersection of PH and AFT. Proportional odds models (PO) - The intersection of PO and AFT - Recidivism in the U.S. 

Practical Assignments:

1.Lab exercise on parametric modelling pf survival data

2.Lab exercise on the proportional hazard model

3.Lab exercise on AFT models

One-sample estimation with censored data - The Kaplan-Meier estimator - Greenwood's formula - The Nelson-Aalen estimator - The expectation of life - Comparison of several groups: Mantel- Haenszel and the log-rank test. 

Regression: Cox's model and partial likelihood - The score and information - The problem of ties - Tests of hypotheses - Time-varying covariates - Estimating the baseline survival - Martingale residuals.

Practical Assignments:

7.Lab exercise on Kaplan-Meier estimator and Nelson-Aalen estimator

8.Lab exercise on Mantel- Haenszel and the log-rank test

9.Lab exercise on the Cox model with time-varying covariate

Cox's discrete logistic model and logistic regression - Modelling grouped continuous data and the complementary log-log transformation - Piece-wise constant hazards and Poisson regression - Current status data versus retrospective data - Open intervals and time since the last event - Backward recurrence times - Interval censoring. 

Practical Assignments:

10.Lab exercise on the discrete logistic model for survival data

11.Lab exercise on Poisson regression for survival data

12.Lab exercise on Piece-wise regression for survival data

Modelling multiple causes of failure - Research questions of interest - Cause-specific hazards - Overall survival - Cause-specific densities - Estimation: one-sample and the generalized Kaplan- Meier and Nelson-Aalen estimators - The Incidence function - Regression models - Weibull regression - Cox regression and partial likelihood - Piece-wise exponential survival and multinomial logits - The identification problem - Multivariate and marginal survival - The Fine-Gray model.

Practical Assignments:

13.Lab exercise on non-parametric modelling of competing risk data 

14.Lab exercise on parametric modelling of competing risk data

15.Lab exercise on multivariate survival data

2. Cleves, M.; W. G. Gould, and J. Marchenko (2016). An Introduction to Survival Analysis using Stata. Revised 3rd Ed. College Station, Texas: Stata Press. 

3. Kalbfleisch, J. D., & Prentice, R. L. (2011). The statistical analysis of failure time data,2nd Ed. John Wiley & Sons. 

4. Moore, D. F. (2016). Applied survival analysis using R. Switzerland: Springer. 

2. Therneau, T. M. and P. M. Grambsch (2000). Modelling Survival Data: Extending the Cox Model, Springer, NY

3. Collett, D. (2015). Modelling survival data in medical research. Chapman and Hall/CRC. 

ESE - 50%

This course is designed to train the students to develop their modelling skills in mathematics through various methods of optimization. The course helps the students to understand the theory of optimization methods and algorithms developed for solving various types of optimization problems. 

Introduction to optimization – convex set and convex functions – simplex method: iterative nature of simplex method – additional simplex method: duality concept - dual simplex method – generalized simplex algorithm - revised simplex method: revised simplex algorithm – development of the optimality and feasibility conditions.   

Practical Assignments:

1. Formulate the LPP.

2. Solve the LPP using simplex method.

3. Solve the LPP using revised simplex method.                                                                                                                  

Branch and bound algorithm – cutting plane algorithm – transportation problem: north-west method, least-cost method, vogel’s approximation and method of multipliers – assignment problem: mathematical statement, Hungarian method, variations of assignment problems.

 Practical Assignments:

4. Solve integer LPP by cutting plane method.

5. Formulate and solve transportation problems.

6. Formulate and solve assignment problems.  

Introduction – unimodal function – one dimensional optimization: Fibonacci method – golden Section Method – quadratic interpolation methods - cubic interpolation methods – direct root method: newton method and quasi newton method – Multidimensional unconstrained optimization: univariate method – Hooks and Jeeves method – Fletcher – Reeves method - Newton’s method and quasi newton’s method.

Practical Assignments:

7. Solve a non LPP problem.

8. Solve an unconstrained optimization problem by a univariate method

Single variable optimization – multivariable optimization with no constraints: semi-definite case and saddle point – multivariable optimization with equality constraints: direct substitution – method of constrained variation – method of Lagrange multipliers - Kuhn-Tucker conditions - constraint qualification – convex programming problem.

Practical Assignments:

9. Solve a single variable optimization problem.

10. Solve multivariable optimization problems with equality constraints.

11. Solve a  convex optimization problem.

Unconstrained minimization problem – solution of an unconstrained geometric programming problem using arithmetic-geometric inequality method – primal dual relationship - constrained minimization - dynamic programming: Dynamic programming algorithm – solution of linear programming problem by dynamic programming.

 Practical Assignments:

12. Formulate and solve a dynamic programming problem.

13. Solve LPP through dynamic programming problems.

14. Solve a  geometric programming problem.

ESE-50%

The course will be inculcating research culture which will enhance the employability skills to the students. 

Students will do the following,

1. Identify a domain for the research project

2. Literature survey 

3. Identifying the existing methodology and models

4. Writing a problem statement 

5. Project presentation at the end of the process

ESE - 50%

Robust estimation: The influence curve and empirical influence curve - M-estimation: Median, Trimmed and winsorized mean - Influence curve for M-estimators -  Limiting distribution of M-estimators - Resampling methods: Quenouille’s Jackknife estimation, parametric and non-parametric bootstrap methods.

Basic concepts in statistical hypotheses testing - Simple and composite hypothesis - Critical regions - Type-I and Type-II error - Significance level - p-value and power of a test - Randomised and non-randomized tests - Neyman-Pearson lemma and its applications - Generalization of NP lemma - Construction of tests using NP lemma - Most powerful test - Uniformly most powerful test - Monotone Likelihood Ratio (MLR) property - Testing in one-parameter exponential families - Unbiased and invariant tests - Locally most powerful tests.

One-sided uniformly most powerful tests - Unbiased and Uniformly Most Powerful Unbiased tests for different two-sided hypothesis - Extension of these results to Pitman family when only upper or lower end depends on the parameters - UMP test from α-similar tests and α-similar tests with Neyman structure.

Likelihood ratio test (LRT) - asymptotic properties - LRT for the parameters of binomial and normal distributions - Generalized likelihood ratio tests - Chi-Square tests - t-tests - F-tests - Need for sequential tests - Sequential Probability Ratio Test (SPRT) - Wald’s fundamental identity - OC and ASN functions - Applications to Binomial, Poisson, and Normal distributions.

Non-parametric tests: Sign test - Chi-square tests - Kolmogorov-Smirnov one sample and two samples tests - Median test - Wilcoxon Signed Rank test - Mann- Whitney U-test - Test for Randomness - Runs up and runs down test - Wald–Wolfowitz run test for equality of distributions - Kruskal–Wallis one-way analysis of variance - Friedman’s two-way analysis of variance - Power and asymptotic relative efficiency.

1. Rohatgi, V. K. and Saleh, A.K.M. (2015). An Introduction to Probability and

Statistics, John Wiley and Sons.

2.  Lehmann, E. L., and Romano, J. P. (2005). Testing Statistical Hypotheses, 2/e, John Wiley, New York.

2nd edition. Mc-Millan, New York.

4. Kale, B. K. and Muralidharan, K.  (2015). Parametric Inference: An Introduction. Alpha Science Int. Ltd.
5. Mukhopadhyay, P.(2015): Mathematical Statistics, Books and Allied (P) Ltd., Kolkata.
6. Gibbons J.K. (1971).  Non-Parametric Statistical Inference, McGraw Hill.

The exposure provided to the multivariate data structure, multinomial and multivariate normal distribution, estimation and testing of parameters, various data reduction methods would help the students in having a better understanding of research data, its presentation and analysis. This course helps to understand multivariate data analysis methods and their applications in various research areas.

Basic concepts on multivariate variables - Multivariate normal distribution - Marginal and conditional distribution - Concept of random vector - Its expectation and Variance - Covariance matrix. Marginal and joint distributions - Conditional distributions and Independence of random vectors - Multinomial distribution - Characteristic functions in higher dimensions - Multiple regressions and multiple correlations -Partial regression and Partial correlation (illustrative examples).

Multivariate analysis of variance (MANOVA) and Covariance (MANCOVA) of one and two-way classified data with their interactions - Univariate and Multivariate Two-Way Fixed-effects Model with Interaction.

Wishart distribution (definition, properties) -Construction of tests -Union - Intersection and likelihood ratio principles - Inference on mean vector - Hotelling's T²- Comparing Mean Vectors from Two Populations - Bartlett’s Test.

Concepts of discriminant analysis - Computation of linear discriminant function (LDF) - Classification between k multivariate normal populations based on LDF - Fisher’s Method for discriminating two or several populations - Evaluating Classification Functions - Probabilities of misclassification and their estimation - Mahalanobis D2.

Factor analysis: - Orthogonal factor model -Factor loadings -Estimation of factor loadings -Factor scores and Its applications.

Cluster Analysis: - Distances and similarity measures - Hierarchical clustering methods - K- Means method.

Wiley. New York.

2. Johnson, R.A. and Wichern, D.W. (2018). Applied Multivariate Statistical Analysis.

6th edn. Prentice-Hall. London.

Mathematical Statistics. 2nd edn. John Wiley & Sons. New York.

2. Srivastava, M.S. and Khatri, C.G. (1979). An Introduction to Multivariate Statistics.

North-Holland.

3. Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. John Wiley. New

York.

The course considers statistical techniques to evaluate processes occurring through time. It introduces students to time series methods and the applications of these methods to different types of data in various fields. Time series modeling techniques including AR, MA, ARMA, ARIMA, and SARIMA will be considered with reference to their use in forecasting. The objective of this course is to equip students with various forecasting techniques and to familiarize themselves with modern statistical methods for analyzing time-series data.

Stochastic Process - Time series as a discrete parameter stochastic process - Auto – Covariance - Autocorrelation and their properties - Exploratory time series analysis- graphical analysis - classical decomposition model - concepts of trend, seasonality and cycle - Estimation of trend and seasonal components-Elimination of trend and seasonality - Method of differencing - Moving average smoothing - Method of seasonal differencing

Practical Assignments:

1.Graphical representation of time series, plots of ACF and PACF and their interpretation

2.Examples of  trend, seasonal and cyclical time series and estimation of trend and seasonal  components

3. Exercise on Moving average smoothing to eliminate trend and illustration on the method of differencing to eliminate trend and seasonality.

4.Exercise on least-square fitting to estimate and eliminate the trend component.

Stationary time series models - Concepts of weak and strong stationarity - General linear Process - Auto-Regressive(AR), Moving Average(MA), and Auto-Regressive Moving Average (ARMA) processes – their properties - conditions for stationarity and invertibility -model identification based on ACF and PACF- Maximum likelihood estimation - Yule Walker Estimation - order selection ( AIC and BIC ) - Residual Analysis- - Box Jenkins methodology to the identification of stationary time series models

Practical Assignments:

5.Exercise on fitting AR model

6. Exercise on fitting MA model

7. Exercise on fitting ARMA model

8.Model-identification using ACF and PACF, Model selection using AIC and BIC

9. Residual analysis and diagnosis check for AR, MA, and ARMA models

Concept of non-stationarity - Spurious trends and regressions-unit root tests: Dickey-Fuller (DF) test - Augmented Dickey-Fuller(ADF) test – Auto-Regressive Integrated Moving Average(ARIMA(p,d,q)) models - Difference equation form of ARIMA- Random shock form of ARIMA - An inverted form of ARIMA

 Practical Assignments: 

10. Exercise on the identification of non-stationary series from various plots.

11. Exercise on testing non-stationarity using ADF test, Exercise on fitting ARIMA models.

12. Residual analysis and diagnosis check for the ARIMA model.

Analysis of seasonal models - parsimonious models for seasonal time series - Seasonal unit root test (HEGY test) - General multiplicative seasonal models - Seasonal ARIMA models - estimation - Residual analysis for seasonal time series.

Practical Assignments:

13. Exercise on the identification of additive and Multiplicative time series

14.Exercise on testing the presence of seasonality and on fitting Seasonal ARIMA models

15. Residual analysis and diagnosis check for Seasonal ARIMA model

In sample and out of sample forecast - Simple exponential and moving average smoothing - Holt Exponential Smoothing - Winter exponential smoothing - Forecasting trend and seasonality in Box Jenkins model:- Method of minimum mean squared error(MMSE) forecast - their properties - forecast error

Practical Assignments:

16.Exercise on Simple exponential smoothing and Holt Exponential Smoothing

17.Exercise on Winters exponential smoothing.

18.Exercise on forecasting using ARIMA models.

19.Exercise on forecasting using seasonal ARIMA models.

2. Chatfield, C., & Xing, H. (2019). The analysis of time series: an introduction with R. CRC Press.

2. Brockwell, P. J., & Davis, R. A. (2016). Introduction to time series and forecasting. springer.

ESE - 50%

Machine learning has a wide array of applications that belongs to different fields, such as biomedical research, reliability of large structures, space research, digital marketing, etc. This course will equip students with a wide variety of models and algorithms for machine learning and prepare students for research or industry application of machine learning techniques. 

Statistical learning: definition-prediction accuracy and model interpretability-supervised and unsupervised learning-assessing model accuracy- important problems in data mining: classification, regression, clustering, ranking, density estimation- Concepts: training and testing, cross-validation, overfitting, bias/variance tradeoff, regularized learning equation- simple and multiple linear regression algorithms.

Practical Assignments:

1.     Lab exercise on data preparation and using simple linear regression

2.     Lab exercise on model assessment simple linear regression

3.     Lab exercise on data preparation with multiple linear regression

Logistic model- training and testing the model-linear discriminant analysis-quadratic discriminant analysis- Use of Bayes’ theorem-k- nearest neighbours - Naive Bayes’- Adaboost.

Practical Assignments:

4.     Lab exercise on the logistic model

5.     Lab exercise on discriminant analysis

6.     Lab exercise on  Naïve Bayes’ and k-NN classifiers

Optimal model-shrinkage methods: ridge and lasso regression-Dimension reduction methods: principal component (PC) regression and partial least square (PLS) regression: Non-linear models: regression splines-polynomial – Generalized additive models

Practical Assignments:

7.     Lab exercise on ridge regression

8.     Lab exercise on Lasso regression

9.     Lab exercise on PC regression

10.     Lab exercise on PLS regression

Decision tree-regression trees - bagging - random forests - boosting - classification trees-boosting-tree vs linear models.

Practical Assignments:

11.     Lab exercise on decision trees

12.     Lab exercise on regression trees

13.     Lab exercise on random forests

14.     Lab exercise on classification trees

Maximal classifier-support vector classifiers-support - rank boost (ranking algorithm) - hierarchical Bayesian modelling for density - resampling techniques-bootstrap- clustering algorithms: K-means algorithm.

Practical Assignments:

15.     Lab exercise on SVM classifier

16.     Lab exercise on rank boost algorithm

17.     Lab exercise on kernel density estimation

18.     Lab exercise on k-means clustering

ESE - 50%

This course provides an understanding of various statistical methods in describing and analyzing biological data. Students will be equipped with an idea about the applications of statistical hypothesis testing,  related concepts and interpretation in biological data.

Presentation of data - graphical and numerical representations of data - Types of variables, measures of location - dispersion and correlation - inferential statistics - probability and distributions - Binomial, Poisson, Negative Binomial, Hyper geometric and normal distribution.

Practical Assignments:

1. Exercise on the representation of data
2. Exercise on reporting data by descriptive statistics

Parametric methods - one sample t-test -  independent sample t-test - paired sample t-test - one-way analysis of variance - two-way analysis of variance - analysis of covariance - repeated measures of analysis of variance - Pearson correlation coefficient - Non-parametric methods: Chi-square test of independence and goodness of fit - Mann Whitney U test - Wilcoxon signed-rank test - Kruskal Wallis test - Friedman’s test - Spearman’s correlation test.

Practical Assignments:

1. Exercise on various parametric methods of analysis
2. Exercise on various non-parametric methods of analysis

Review of simple and multiple linear regression - introduction to generalized linear models - parameter estimation of generalized linear models - models with different link functions   - binary (logistic) regression - estimation and model fitting - Poisson regression for count data -  mixed effect models and hierarchical models with practical examples.

Practical Assignments:

1. Exercise on simple linear and multiple linear regression
2. Exercise on logistic regression
3. Exercise on Poisson regression

Introduction to epidemiology, measures of epidemiology, observational study designs: case report, case series correlational studies, cross-sectional studies, retrospective and prospective studies, analytical epidemiological studies-case control study and cohort study, odds ratio, relative risk, the bias in epidemiological studies.

Practical Assignments:

1. Exercise on analysis of observational study data
2. Exercise on analysis of cross-sectional study data
3. Exercise on analysis of case-control study data
4. Exercise on analysis of cohort study data

Introduction to demography, mortality and life tables, infant mortality rate, standardized death rates, life tables, fertility, crude and specific rates, migration-definition and concepts population growth, measurement of population growth-arithmetic, geometric and exponential, population projection and estimation, different methods of population projection,  logistic curve, urban population growth, components of urban population growth.

Practical Assignments:

1. Exercise on preparing life tables
2. Exercise on measures of population growth

50% End Semester Examination.

This course will provide knowledge in different probability models in the reliability evaluation of the system and its components. Reliability engineering is applied in the industry to reduce failures, ensure effective maintenance and optimize repair time.

Reliability of a system - failure rate - mean, variance and percentile residual life: identities connecting them - notions of ageing - IFR, IFRA, NBU, NBUE, DMRL, HNBUE, NBUC, etc. and their mutual implications - TTT transforms and characterization of ageing classes.

Practical Assignments:

1. Exercise on failure rate function and mrl function
2. Exercise on comparison ageing classes

Non-monotonic failure rates and mean residual life functions - study of lifetime models: exponential, Weibull, lognormal, generalized Pareto, gamma with reference to basic concepts and ageing characteristics - bathtub and upside-down bathtub failure rate distributions

Practical Assignments:

      3. Exercise on exponential lifetime model

      4. Exercise on Weibull lifetime model

      5. Exercise on bathtub shaped lifetime model 

Reliability systems with dependents components: Parallel and series systems, k out of n Systems - ageing properties with dependent and independents components -  concepts and measures of dependence on reliability - RCSI, LCSD, PF2, WPQD.

Practical Assignments:

6. Exercise on reliability evaluation of series system

7. Exercise on reliability evaluation of a parallel system

8. Exercise on reliability evaluation of k out of n system

9. Exercise on reliability evaluation of dependent component system

Reliability estimation using MLE: exponential, Weibull and gamma distributions based on censored and non-censored samples - UMVU estimation of reliability function - Bayesian reliability estimation of exponential and Weibull models

Practical Assignments:

10. Exercise on ML estimation under non-censored samples.

11. Exercise on ML estimation under censored samples.

12. Exercise on  Bayesian estimation of reliability.

Life testing: basics – modelling lifetime – Accelerated Life Time (ALT) models- cumulative exposure models (CEM) - exponential CEM – stress-strength reliability – exponential stress-strength model.

Practical Assignments:

13. Exercise on basic life testing procedure.

14. Exercise on exponential CEM model.

15. Exercise on stress-strength reliability.

2.      Bain, L. (2017). Statistical analysis of reliability and life-testing models: theory and methods. Routledge.

ESE - 50%

This course deals with the theory and application of different numerical methods techniques to solve the complex problems that arise in the modern world of science. The course highlights that through the numerical algorithms, it is definite to arrive at a solution which is efficient and stable for large scale systems.  

Errors and their analysis – Floating Point representation of numbers – Solution of algebraic and Transcendental equations: Bisection method, fixed-point iteration method, the method of False position, Newton Raphson method and Muller’s method. The solution of linear systems – Matrix inversion method – Gauss elimination method – Gauss-Seidel and Gauss-Jacobi iterative methods.

Practical Assignments:

1. Solutions to algebraic equations using Bisection and fixed point methods

2. Solutions to algebraic and transcendental equations using Newton Raphson and Muller’s method.

3. Solving system of linear equations using Matrix inversion and Gauss elimination methods

4. Finding real roots to a system of linear equations using Gauss-Seidel and Gauss-Jacobi iterative methods.

Convergence criterion, Aitken’s-process - Sturm sequence method to identify the number of real roots, Bairstow’s method - Graeffe’s root squaring method - Birge-Vieta method - Solution of Linear system of algebraic equations: LU-decomposition methods (Crout’s, Cholesky and Delittle methods), consistency and ill-conditioned system of equations, Tridiagonal system of equations, Thomas algorithm. 

Practical Assignments:

5. Identifying real roots for algebraic equations using the Sturm sequence method and Bairstow’s method.

6. Solving equations using the Birge-Vieta method.

7. Solving system of linear equations using LU decomposition methods.

8. Examining consistency of the system of equations using Tridiagonal system of equations and Thomas algorithm. 

Finite difference: Forward difference, Backward difference and Shift operators – Separation of symbols – Newton’s Formula for interpolation – Lagrange’s interpolation formulae – Numerical differentiation – Numerical integration: Trapezoidal rule, Simpson’s one-third rule and Simpson’s three-eight rule. Numerical ODE: Taylor’s series – Picard’s method – Euler’s method – Modified Euler’s method – Runge Kutta Method.

Practical Assignments:

9. Exercise on Newton’s and Lagrange's interpolation formulae

10. Integration using Trapezoidal rule and Simpon’s rules.

11. Numerical Solutions for ODE using Taylor’s series and Picard’s Method

12. Numerical Solutions for ODE using Euler’s method, Modified Euler’s method and Runge Kutta Method.

Lagrange, Hermite, Cubic-spline’s method – with uniqueness and error term, polynomial interpolation: Chebychev and Rational function approximation, Gaussian quadrature, Gauss-Legendre, Gauss-Chebychev formulas.  

Practical Assignments:

13. Integration using Hermite and Cubic-spline’s method

14. Interpolation for polynomial equations Chebychev and rational function approximation

15. Interpolation for polynomial equations Gaussian quadrature and Gauss-Legender

16. Numerical integration through Gauss-Chebyshev formulas.

Initial value problems – Multistep method – Adams-Moulton method – Stability (convergence and truncation error) – Boundary value problems: second order finite difference method – first, second and third types by shooting method – Rayleigh-Ritz method – Galerkin method.  

Practical Assignments:

17. Solution for initial value problems using Multistep and Adams-Moulton method

18. Solving ODE using shooting, Rayleigh-Ritz and Galerkin methods

2.      S.C. Chopra and P.C. Raymond, Numerical Methods for Engineers, New Delhi: Tata McGraw-Hill, 2010.

3.      Graham. W Griffiths, Numerical Analysis using R solution to ODEs and PDEs, Cambridge University Press, 2016.

4.      Jaan Kiusalaas, Numerical methods in Engineering with Python 3, Cambridge University Press, 2013.

ESE- 50%

This course will provide the basic theory and computing tools to perform nonparametric tests, including the Sign test, Wilcoxon signed-rank test, Median test etc. Kruskal-Wallis for one-way and multiple comparisons, linear rank test for location and scale parameters and measure of association in bivariate populations are other nonparametric tests covered in this course. The aim of the course is the in-depth presentation and analysis of the most common methods and techniques of non-parametric statistics such as sign test, rank test, run test, median test etc.

 The quantile function - the empirical distribution function - statistical properties of order statistics- confidence interval for a population quantile -hypothesis testing for a population quantile -the sign test and confidence interval for the median - rank-order statistics -treatment of ties in rank tests-  Wilcoxon signed-rank test and confidence interval

Practical Assignments:

1. Exercise on confidence interval estimation and hypothesis test for a population Quantile

2. Exercise on sign test and confidence interval for the median

3. Exercise on rank-order statistics -treatment of ties in rank tests

4. Exercise on Wilcoxon signed-rank test and confidence interval

Wald-Wolfowitz runs test - Kolmogorov-Smirnov two-sample test - median test - the control median test - the Mann-Whitney U test                                                                                    

Practical Assignments:

 5.Exercise on Wald-Wolfowitz runs test

6.Exercise on Kolmogorov-Smirnov two-sample test.

7.Exercise on Median test and control median test.

8.Exercise on Mann-Whitney U test.

Definition of linear rank statistics - Wilcoxon rank-sum test - mood test - Freund-Ansari-Bradley-David-Barton tests - Siegel-Tukey test

Practical Assignments:

9.Exercise on Wilcoxon rank-sum test and  mood test

10.Exercise on Freund-Ansari-Bradley-David-Barton tests.

11. Exercise on Siegel-Tukey test

extension of the median test - the extension of the control median test - the Kruskal-Wallis one-way ANOVA test and multiple comparisons - tests against ordered alternatives - comparisons with a control -  Chi-Square test for k proportions

Practical Assignments:

12.Exercise on the extension of the median test and control median test.

13.Exercise on Kruskal-Wallis one-way ANOVA test.

14.Exercise on chi-square test for k  proportions

Introduction: definition of measures of association in a bivariate population - Kendall’s Tau coefficient - Spearman’s coefficient of rank correlation - relations between R and T; E(R), t, and r 

Practical Assignments:

15.Exercise on Kendall’s Tau coefficient.

16.Exercise on Spearman’s coefficient of rank correlation

1. Hollander, M., Wolfe, D. A., & Chicken, E. (2013). Nonparametric statistical methods (Vol. 751). John Wiley & Sons.

2. Lewis, N. D. C. (2013). 100 Statistical Tests in R. Heather Hills Press.

ESE 50%

Game theory and the decision is a branch of Mathematics and Statistics that enables to study of the strategic interactions amongst rational decision-makers. Traditionally, game-theoretic tools have been applied to solve problems in Economics, Business, Political Science, Biology, Sociology, Computer Science, Logic, and Ethics. In recent years, applications of game theory have been successfully extended to several areas of engineered / networked system such as wireline and wireless communications, static and dynamic spectrum auction, social and economic networks.

Theory of games - zero-sum game - minimax - maxmin -  dominance strategy - the value of the game - Basic elements of game and Decision -  Comparison of the two theories - Decision function and Risk function; Randomization and optimal decision rules - Form of Bayes rules for estimation.                                                                                                         

Practical Assignments:

1. Two-person zero-sum game
2. Game with minimax and maximin strategies
3. Game with dominance rule and value of the game.

Admissibility and completeness - Fundamental theorems of Game and Decision theories - Admissibility of Bayes rules - Existence of Bayes decision rules - Existence of minimal complete class - Essential completeness of the class of non-randomized decision rules - Minimax theorem -  The complete class theorem -  Methods for finding minimax rules.                   

Practical Assignments:

1. Admissibility of Bayes rules
2. Examining the completeness
3. Problems on the non-randomized decision rule

Sufficient Statistics and essentially complete class of rules based on Sufficient Statistics - Complete Sufficient Statistics - Continuity of the risk function.         

Practical Assignments:

1. Examining the complete sufficient statistic
2. Rules-based on sufficient statistic

Invariant decision problems and rules - Admissible and minimax invariant rules -  Minimax estimates of location parameter - Minimax estimates for the parameters of normal distribution.                                                                                                        

Practical Assignments:

1. Minimax estimates of Location Parameter
2. Minimax estimates for the normal parameters.
3. Invariant minimax rules

Monotone Multiple decision problems - Bayes rules in multiple decision problems -  Slippage problems.

Practical Assignments:

1. Bayes rules in multiple decision problems
2. Slippage problems

    .

1.  Robert Winkler (Jan 15, 2003) An Introduction to Bayesian Inference and Decision, Second  Edition.
2. T.S. Ferguson (1967), Mathematical Statistics – A Decision Theoretic, Approach,  Academic Press.

3. M.H. DeGroot (1976), Optimal Statistical Decisions, McGraw Hill.

ESE - 50%

This will equip the student to apply statistical methods they have studied in various courses and present their work through research articles.

1. Apply various statistical methods in solving a real-life problem.

2. Comparison with the existing models or results.

3. Writing research article 

4. Presentation of the article

ESE 50%

Operations research helps in solving problems in different environments that need decisions. The module includes the topics : linear programming, integer programming, nonlinear programming, simple queueing models and inventory models. The aim of the course is to provide the students, how to formulate the problems into mathematical models and to use the appropriate methods to solve them. 

General Linear programming problem - Formulation -  Solution through graphical, Simplex, Big-M and Two phase methods - Revised Simplex method - Big-M and Two phase Revised Simplex methods - Duality - Primal-dual relationships - Dual Simplex method.

Gomory’s All-Integer Cutting-Plane Method - Construction of Gomory’s Constraint - Gomory’s Mixed-Integer Cutting-Plane Method -  Construction of Additional Constraint for Mixed-Integer Programming Problem - Branch and Bound Method.

General nonlinear programming problem - Constrained optimization with equality constraints - Necessary conditions for a generalized NLPP (without proof) - Sufficient conditions for a general NLPP with one constraint (without proof) - Sufficient conditions for a general problem with m(<n) constraints (without proof) -Constrained optimization with inequality constraints - Kuhn-Tucker conditions for general NLPP with m(<n) constraints (without proof).  Constrained optimization with inequality constraints - Kuhn-Tucker conditions for general NLPP with m(<n) constraints (without proof)

Basics of queuing model - Probability distribution in a queueing system - Distribution of arrivals (Pure birth model) - Distribution of departures (Pure death model) -  Poisson queuing model:  (M/M/1) : (GD/∞/∞) - (M/M/1) : (N/FCFS/∞) - (M/M/c) : (∞/FCFS/∞) - (M/M/c) : (N/FCFS/∞).

Deterministic inventory Models - Economic Order Quantity(EOQ) models - Classic EOQ models - Problems with no shortages - The fundamental EOQ Problems: EOQ problems with several production runs of unequal length - Problems with price breaks - One price break - More than one price break - Probabilistic inventory models - Single Period Problem without set-up cost - I.

2. Taha, H. A. (2013). Operations research: an introduction. Pearson Education India.

3. Shortle, J. F., Thompson, J. M., Gross, D., & Harris, C. M. (2018). Fundamentals of queueing theory (Vol. 399). John Wiley & Sons.

4. Sharma, J. K. (2016). Operations research: theory and applications. Trinity Press, an imprint of Laxmi Publications Pvt. Limited.

This course will provide students with a mathematical background of various basic designs involving one-way and two-way elimination of heterogeneity and characterization properties. To prepare the students in deriving the expressions for analysis of experimental data and selection of appropriate designs in planning a scientific experimentation

Basic principles of design of experiments - Randomization - Replication and Local control - Uniformity trials - Size and Shape of plots and blocks - Elements of linear estimation - Analysis of variance - Completely Randomized Design (CRD) - Randomized Complete Block Design (RCBD) and Latin Square Design (LSD) -  Missing plot techniques

Analysis of covariance - Ancillary/Concomitant variable and study variable - Linear model for ANCOVA - Adjustment of treatment sum of squares in ANCOVA - One - Way and two-way classification with a single concomitant variable in CRD and RCBD designs.

Factorial experiments - Simple experiment (single factor) vs Factorial experiments - Mixed and Fixed factor experiments - Treatment combination in a factorial experiment - Simple effect - Main effect and Interaction effect in a factorial experiment - Yates method of computing factorial effects totals - Complete and partial confounding in symmetrical factorial experiments (2², 2³, 3³, 2ⁿand 3ⁿ series) - Gain in the factorial experiments.

Split - Plot, Split - Split plot and Strip - Plot (Split Block) design - Situation for the usage of the design - Layout and analysis of the designs - Difference in the error components in the designs - Selection of factor for allocation in plots (main/sub) - Combined experiments -  Cross - Over designs

Balanced Incomplete Block (BIB) designs - General properties and Analysis with and without recovery of information - Construction of BIB designs - Parameter relationship - Intra and inter-block Analysis - Partially Balanced Incomplete Block Design (PBIBD) - Youden square designs - Lattice designs

This course provides an introduction to the application of statistical tools in the industrial environment to study, analyze and control the quality of products

Meaning and scope of statistical quality control - Causes of quality variation - Control charts for variables and attributes - Rational subgroups - Construction and operation of, σ, R, np, p, c and u charts - Operating characteristic curves of control charts. Process capability analysis using histogram, probability plotting and control chart - Process capability ratios and their interpretations.

Specification limits and tolerance limits - Modified control charts - Basic principles and design of cumulative - sum control charts – Concept of V-mask procedure – Tabular CUSUM charts - Construction of Moving range - moving-average and geometric moving-average control charts.

Acceptance sampling: Sampling inspection by attributes – single, double and multiple sampling plans – Rectifying Inspection - Measures of performance: OC, ASN, ATI and AOQ functions - Concepts of AQL, LTPD and IQL - Dodge – Romig and MIL-STD-105D tables

Sampling inspection by variables - known and unknown sigma variables sampling plan - Merits and limitations of variables sampling plan - single, double and multiple sampling plans - Derivation of OC curve – determination of plan parameters.

Continuous Sampling Plans (CSP): CSP-1- CSP-2 - CSP-3 - Skip-Lot Sampling Plans (SkSP): SkSP-1 - SkSP-2  with SSP as reference plan - Chain Sampling Plans (ChSP - 1) with SSP as reference plan - Tighten-Normal-Tighted (TNT) sampling plan with SSP as reference plan– Decision Lines.

2. Schilling, E. G., and Nuebauer, D.V. (2009). Acceptance Sampling in Quality Control, Second Edition, CRC Press, New York

3. Duncan, A. J. (2003.). Quality Control and Industrial Statistics, Irwin-Illinois, US.

The objective of this course is to provide fundamental knowledge of neural networks and deep learning. This course gives a brief idea of the basics of neural networks, shallow and deep neural networks and other methods to build various research projects.

Fundamental concepts of Artificial Neural Networks (ANN) - Biological neural networks - Comparison between biological neuron and artificial neuron - Evolution of neural networks - Scope and limitations of ANN - Basic models of ANN - Learning methods - Activation functions - Important terminologies of ANN: Weights - Bias - Threshold - Learning Rate - Momentum factor - Vigilance parameters.

Practical  Assignments:

1. Exercise on construction of Artificial Neural Networks.
2. Exercise on training and testing the data 

Concept of supervised learning algorithms - Perceptron networks - Adaptive linear neuron (Adaline) - Multiple adaptive linear neuron - Back-Propagation network: Learning factors - Initial weights - Learning rate ɑ - Momentum factor - Generalization - Training and testing of the data.

Practical  Assignments: 

3. Exercise on multiple adaptive linear neurons 
4. Exercise on Back-propagation networks

Concept of unsupervised learning algorithms - Fixed weight competitive net: Maxnet - Mexican Hat net - Hamming networks - Kohonen self-organizing feature maps - Learning vector quantization.

Practical  Assignments:

5. Exercise on Maxnet and Mexican Hat net.
6. Exercise on Hamming networks
7. Exercise on Kohonen self-organizing feature maps
8. Exercise on Learning vector quantization

Introduction - Components of Convolution Neural Networks (CNN) architecture: Padding - Strides - Rectified linear unit layer - Exponential linear unit - Pooling - Fully connected layers  - Local response normalization - Hierarchical feature engineering -  Training CNN using Backpropagation through convolutions - Case studies: AlexNet - GoogLeNet.

Practical  Assignments:

9. Exercise on building CNN with the rectified linear unit and exponential linear unit
10. Exercise on Hierarchical feature engineering
11. Exercise on training CNN using Backpropagation through convolutions
12. Exercise on feature prediction using AlexNet and GoogLeNet.

Stateless algorithms: Naive algorithms - Upper bounding methods - Simple reinforcement learning for Tic-Tac-Toe - Straw-Man algorithms - Bootstrapping for value function learning - One policy versus off policy methods: SARSA - Policy gradient methods: Finite difference method - Likelihood ratio method - Monte Carlo tree search.

Practical  Assignments:

13. Exercise on Naive and upper bounding algorithms for data classification.
14. Exercise on Bootstrapping for value function learning
15. Exercise on SARSA method
16. Exercise on Finite difference and likelihood ratio methods
17. Exercise on Monte Carlo tree search method.

ESE - 50%